# Questions tagged [forcing]

Forcing is a set theoretic method used mainly for proving independence results. For questions about forcing function please use (differential-equations).

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### "Correct" definition of $\kappa$-closed poset

Compare the following two definitions: (1) a poset is $\kappa$-closed if any decreasing sequence of length $<\kappa$ has a lower bound; (2) a poset is $\kappa$-closed if any decreasing sequence of ...
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### Exercise IV 2.8 in "New Kunen" (Set Theory).

So, I am learning about the most basic setup to get forcing on the ground, and I feel like I want to understand the recursively defined notion \begin{align*} \text{val}(\tau,G) &:= \{\text{val}(\...
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### Is there a structure or abstraction that genuinely generalizes both field extensions and (Cohen) forcing?

I'm just starting with the topic so I don't know much. But I'm always told that the similarity with field extensions is just an analogy and we shouldn't take it so seriously. However, I just must ask ...
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### Consistency Strength of OCA?

I was wondering whether we know the consistency strength of the Open Colouring Axiom (or Todorčević's Axiom). It is a consequence of PFA, so clearly it is below a supercompact, but I cannot find ...
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### What exactly are characteristic functions of sets when constructing Boolean models?

Reading J. L. Bell's Set Theory --- Boolean-Valued Models and Independence Proofs. This is how the introduction of Boolean models is motivated: Suppose that for each set $x ∈ V$ we are given a ...
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### In what sense is forcing "impossible" in $L$?

I just saw an interesting video from Hugh Woodin about Ultimate $L$. In it, he says one of the reasons $L$ is so interesting is because it not only settles many natural set theory questions, but is ...
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### Intuition for working with p-names.

So I have been trying to learn about forcing in set theory from kunen. There are these things called "p-names" which have a complicated definition in forcing. if $M$ is a countable ...
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### Motivation behind Generic filters in forcing.

So I was reading about forcing in wikipedia to try to get an intuitive idea about forcing in set theory. There is this paragraph in it: A subtle point of forcing is that, if $X$ is taken to be an ...
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### How to understand canonical ultrafilter in forcing chapter of Jech

Am new to forcing (obviously...), and I got to read Definition 14.25 in Jech's Set Theory which defines the canonical ultrafilter $\dot{G}$ on the Boolean algebra $B$ which is: $\dot{G}(\check{u})=u$ ...
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### A step in Kunen's proof of ccc preserving cofinalities

I have a question about one of the steps in the proof of "$ccc$ preserves cofinalities" in Kunen. The whole proof is below. My question concerns the claim that $\sup Y=\beta$. I am not sure ...
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### How exactly does $G$ not being generic make $\mathbf{M}[G]$ not a model?

I've been trying to solve this exercise from Nik Weaver's Forcing for Mathematicians. Let $P$ be the set of all finite partial functions from $\mathbb{N}^2$ into $\{0,1\}$, let $α$ be a countable ...
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